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TIN & Surface Interpolation
TRN
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Triangulated Irregular Network (TIN)
• Node (x, y, z)• Edge• Triangles• Topology
– The triangle number – The numbers of each adjacent
triangle – The three nodes defining the
triangle – The x, y coordinates of each node – The surface z value of each node – The edge type of each triangle
edge (hard or soft)
Conversions between Surface Models
ToFrom
Point Triangle Grid
Point Spatial interpolation
TIN generation Spatial interpolation, indirect interpolation
Triangle Edge points Edit TIN Linear / curved surface interpolation
Grid Nearest neighbor, linear/cubic interpolation
TIN generation Nearest neighbor, linear/cubic interpolation
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Static vs. Dynamic Triangulation
• Static: all data are used in a single-pass process
• Dynamic: data are used in a multiple-pass process (i.e., subsequent editing of a TIN)
Ways of Forming Triangles
Basic requirements:1. Unique formation2. Equilateral triangles3. Minimal edge length
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TIN Formation
• Delaunay criterion (empty circumcircle principle): the circumscribing circle of any triangle contains no other points of data except the three defining it.
• Local equiangularity (max-min angle principle): minimizing the range of interior angles in the two triangles that form a convex quadrilateral (through Lawson’s Local Opt. Procedure).
• Minimum sum-distance principle:• Minimum circumscribing circle radius:• …
Static Triangulation of Points
Algorithm:1. Select an initial point2. Search for the first edge3. Form the first triangle (nearest
point to the edge)4. Search for a subsequent edge5. Form a subsequent triangle6. Repeat 4, 5 until all points become
nodes of triangles
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Searching for a Node Point
Searching for a Subsequent Edge
Convex hull algorithm Imaginary bnd algorithm
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Constrained Triangulation
•Random points and feature-specific points of a DTM
•Approaches:–Point densification of
features
–Constrained Delaunay Triangulation
Triangulation from Contours
• Point densification of contours• Contours as soft breaklines
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Triangulation from DEM• Selection of significant points from DEM
– Very Important Points (VIP)– Maximum z-tolerance (d)
• Triangulation from VIP Ph
G P C
Pe
ds
Delineate TIN Data Area –ArcGIS 3D Analyst Tool
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TIN model
TIN model with pixel center points
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TIN model with pixel center points
Spatial Interpolation
• Methods to construct the surface model
• Global vs. local• Exact vs. best fit• Information lost at the sampling stage
can never be reconstructed (?)
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Local MethodNeighbors
Distribution of control pointsExtent of spatial autocorrelation
(a) (b) (c)
(a) find the closest points to the point to be estimated, (b) find points within a radius, and (c) find points within each of the four quadrants.
Distance Weighted Interpolation
•Neighbor points with observations•Weighting functions
s
ki
s
ki
i
d
dz
z
1
1
1
1
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Types of Local Spatial Interpolation
Kriging
Modeled Semivariance
Empirical Semivariance
Semivariance
DistanceNugget
Sill
Range
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DEM Resampling
• Down-scale (Fine to coarse)–Treat output grid as polygons–Data aggregation
• Up-scale (Coarse to fine)–Treat DEM as point data– Interpolation
Spatial Interpolation with Sparse Sample Points
• Convert contours to DEM• Generate DEM from transects
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Contours to DEM
Densification of Sample Points
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Lake Bathymetry
Experiments
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Densification of Sample Points